If we want to be incredibly simplistic, we can look at firearms as a general plus for infantry and ships, and a general minus for cavalry and fortifications, relatively speaking. To be sure, both cavalry and fortification benefit in and of themselves from the introduction of firearms, but the forces that would be used against them benefit more.
With this simplistic view, we can consider the development of firearms to be something of a wash for the Byzantines on land, at least on the eastern frontier. The Arab empires of the time hardly seem like the sort to not catch on to something so useful, and black powder would be inherently more difficult to keep a secret than Greek Fire.
On the other hand, firearms would be much more effective against the less sedentary enemies they faced to their north. Firearms are quite useful against cavalry and, even more importantly, nomadic peoples are generally not all that good at siege warfare, negating significantly the disadvantage to fortifications that gunpowder weaponry brings about, at least on that front. I would expect the Avars, Bulgars, Pechenegs, and/or Cumans to do much less well against the Empire in this situation (depending on when the technology is developed). Having a stronger hand against the nomads could conceivably enable the Empire to maintain a more aggressive policy against the Arabs.
On the other (third?) hand, gunpowder does tend to change the dynamics of the Nomad vs. Farmer conflict throughout history so much that the expansion of the Nomads tends to be stopped pretty well in its tracts, historically, once firearms became widespread (they maximize all the military advantages of sedentary societies and virtually eliminate the advantages of non-sedentary societies). What might very well happen is that, confronted with a gunpowder-armed Byzantine Empire, whatever tribe is attempting to muscle in on them could settle down (possibly even with Byzantine help, as a buffer state to further threats) and prove to be a threat in the long run as they develop their own weaponry.
Lastly (forget any 'hand expressions' now), while it would initially seem that gunpowder would be a boon to the Byzantine navy, its ease of dissemination to other (hostile) powers would likely eliminate the advantage of Greek Fire. Now, I don't know the ranges of early ship cannons, nor do I know the range of Greek Fire syphons, but I have to imagine that cannons, even in their early days had an advantage there (an advantage that would only continue to grow).